
Denote this \(\mathfrak {A}_\mathfrak {R}\) -bimodule by \(\widetilde{\mathcal {M}}\) . To define the cohomology of the \(\lambda \) -weighted Rota-Baxter associative algebra \((\mathfrak {A}, \mathfrak {R})\)  with coefficients in the Rota-Baxter bimodule \((\mathcal {M}, \mathcal {R})\) , the authors in [1]} considered two Hochschild cochain complexes, namely \(\lbrace  C^*_\mathsf {H} (\mathfrak {A}, \mathcal {M}), \delta _\mathsf {H} \rbrace \)  and \(\lbrace  C^*_\mathsf {H} (\mathfrak {A}_\mathfrak {R}, \widetilde{M}), \partial _\mathsf {H} \rbrace \) . The first one is the Hochschild complex of the given algebra \(\mathfrak {A}\)  with coefficients in the bimodule \(\mathcal {M}\) , whereas the second one is the Hochschild complex of the algebra \(\mathfrak {A}_\mathfrak {R}\)  with coefficients in the bimodule \(\widetilde{\mathcal {M}}\) . They proved that the collection \(\lbrace  \Psi ^n : C^n_\mathsf {H} (\mathfrak {A}, \mathcal {M}) \rightarrow C^n_\mathsf {H} (\mathfrak {A}_\mathfrak {R}, \widetilde{M}) \rbrace _{n \ge 0}\)  of maps
given by
\(&\Psi ^0 = \mathrm {id}_\mathcal {M}, ~~ \text{ and } \\&\Psi ^n (f) (a_1, \ldots , a_n) = f (\mathfrak {R}(a_1), \ldots , \mathfrak {R}(a_n)) - \sum _{k=0}^{n-1} \lambda ^{n-k-1} \sum _{i_1 < \cdots < i_k} \mathcal {R} \circ f \big ( a_1, \ldots , \mathfrak {R}(a_{i_1}), \ldots , \mathfrak {R}(a_{i_k}), \ldots , a_n \big )\) 
